| L(s) = 1 | + (−0.248 − 0.968i)2-s + (−2.38 + 0.455i)3-s + (−0.876 + 0.481i)4-s + (2.17 − 0.512i)5-s + (1.03 + 2.20i)6-s + (0.144 − 0.199i)7-s + (0.684 + 0.728i)8-s + (2.70 − 1.07i)9-s + (−1.03 − 1.98i)10-s + (1.25 − 0.322i)11-s + (1.87 − 1.54i)12-s + (−2.25 − 5.69i)13-s + (−0.229 − 0.0907i)14-s + (−4.96 + 2.21i)15-s + (0.535 − 0.844i)16-s + (2.49 − 4.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.175 − 0.684i)2-s + (−1.37 + 0.263i)3-s + (−0.438 + 0.240i)4-s + (0.973 − 0.229i)5-s + (0.422 + 0.898i)6-s + (0.0547 − 0.0753i)7-s + (0.242 + 0.257i)8-s + (0.902 − 0.357i)9-s + (−0.328 − 0.626i)10-s + (0.379 − 0.0973i)11-s + (0.540 − 0.447i)12-s + (−0.625 − 1.58i)13-s + (−0.0612 − 0.0242i)14-s + (−1.28 + 0.572i)15-s + (0.133 − 0.211i)16-s + (0.606 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.497234 - 0.569736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497234 - 0.569736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.248 + 0.968i)T \) |
| 5 | \( 1 + (-2.17 + 0.512i)T \) |
| good | 3 | \( 1 + (2.38 - 0.455i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (-0.144 + 0.199i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 0.322i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (2.25 + 5.69i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-2.49 + 4.54i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.648 + 3.39i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (5.77 + 0.363i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-0.510 - 0.0645i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-6.94 - 3.81i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-4.33 - 2.75i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.483 + 7.68i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (4.67 - 1.52i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (4.93 - 5.25i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-1.67 - 0.786i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-1.54 - 1.86i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.0496 + 0.789i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (1.58 + 12.5i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (6.49 + 6.09i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-8.59 - 7.10i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (1.57 + 8.24i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (-7.97 - 1.52i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (8.61 - 10.4i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (1.20 - 9.54i)T + (-93.9 - 24.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.85516194535099311749246436647, −10.72381809189985690917490771641, −10.13618441732144080346933659786, −9.374332811937390573886799858842, −7.941736046837656587630790940881, −6.46958768174297500116722193044, −5.42791476802287930432279482836, −4.75563498953978958803976073206, −2.82129096559020023890195894198, −0.807889133556870622274844145542,
1.69415077470300210593886693435, 4.31625565894940169480095259265, 5.56509698234598693325380127296, 6.25189023033680758486914347382, 6.91959983849506783764227004401, 8.308562075339351693515386694668, 9.705475428562245125108119505375, 10.19026998964929889108946737568, 11.51469045278733629724237348577, 12.14537565000245219661079251020
